In accordance with the previous works in the field beginning with the one of Burbea [1]}, we should rewrite the equation (REF ) by using conformal mappings. For that purpose we shall now present the function spaces used throughout this first part on periodic solutions. Notice that we shall identify \(2\pi \) -periodic \(g:\mathbb {R}\rightarrow \mathbb {C}\)  functions with functions \(f:\mathbb {T}\rightarrow \mathbb {C}\)  defined on the torus \(\mathbb {T}=\mathbb {R}/2\pi \mathbb {Z}\cong \mathbb {U}\)  through the relation
\(f(w)=g(\theta ),\quad w=e^{{\rm i} \theta }.\) 
